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Proposition 4.8.8.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The functor $F$ is essentially $n$-categorical.

$(2)$

The comparison map $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Remark 4.8.8.15 is an equivalence of $\infty $-categories.

Proof. It follows from Proposition 4.8.8.14 (and Proposition 4.8.6.34) that the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ is essentially $n$-categorical. By virtue of Remark 4.8.6.8, we can replace $(1)$ by the following condition:

$(1')$

The functor $F'$ is essentially $n$-categorical: that is, it is $m$-full for $m \geq n+2$.

Since $F'$ is also $m$-full for $m \leq n+1$ (Corollary 4.8.8.19), the equivalence $(1') \Leftrightarrow (2)$ follows from Remark 4.8.5.11. $\square$